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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">IAlph</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">OAlph</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">tfsm</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_1.html#L2" target="_self">Mealy-FSM</a> of <font color="Maroon">IAlph</font>,<font color="Maroon">OAlph</font>;<br/>







<b>set </b><font color="Maroon">K</font> = <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">K1</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>;<br/>
<b>set </b><font color="Maroon">S</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font>;<br/>
<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a><br/>

<b>assume </b><a NAME="E1:62"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font><b> such that </b><br/><a NAME="E4:62"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="fsm_1.html#E23:62"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K2">proj</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font> ;<br/>
<b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E7:62"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K7" target="_self">-eq_states_partition</a> <font color="Maroon">tfsm</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E8:62"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E9:62"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> ;<br/>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span> <b>by </b><i><a class="txt" href="fsm_1.html#E23:62"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">Q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:62"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E14:62"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">Q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E15:62"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">Q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E16:62"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Q</font>,<font color="Maroon">x'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<a NAME="E17:62"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">x'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/>
<a NAME="E18:62"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">q</font>,<font color="Maroon">x'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:62"/><b>then </b>
<font color="Maroon">k</font> <a href="fsm_1.html#R6" target="_self">-equivalent</a> <font color="Maroon">q</font>,<font color="Maroon">x'</font>
 
<b>by </b><i><a class="txt" href="fsm_1.html#E13:62"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E20:62"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">x'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:62"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">Q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:62"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> &amp; <font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">tfsm</font> <b>by </b><i/>;<br/>
<a NAME="E25:62"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D1">BORSUK_1:def 1</a></i>;<br/>
<a NAME="E26:62"/><b>then </b>
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="fsm_1.html#K6" target="_self">-eq_states_EqR</a> <font color="Maroon">tfsm</font></span>)</span>,<font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E27:62"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>


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